Mathematics Questions and Answers

Part-A

Section – I

1. Find H.C.F. and L.C.M. of 220 and 284 by Prime factorisation method.

Prime Factorisation of 220:

220 ÷ 2 = 110

110 ÷ 2 = 55

55 ÷ 5 = 11

11 ÷ 11 = 1

Prime Factorisation of 220: 220 = 22 × 5 × 11

Prime Factorisation of 284:

284 ÷ 2 = 142

142 ÷ 2 = 71

71 ÷ 71 = 1

Prime Factorisation of 284: 284 = 22 × 71

H.C.F.: The common factor is 22, so HCF = 4.

L.C.M.: The LCM is 22 × 5 × 11 × 71 = 1540.

2. Check whether A = { x : x2 = 25 and 6x = 15 } is an empty set or not? Justify your answer.

Condition 1: x2 = 25

The possible values of x are 5 and -5.

Condition 2: 6x = 15

Solving for x: x = 2.5.

Since there is no common value between x = 5 or -5 (from condition 1) and x = 2.5 (from condition 2), the set is empty.

3. The sum of zeroes of a quadratic polynomial Kx2 – 3x + 1 is 1. Find the value of K.

The sum of zeroes of a quadratic polynomial ax2 + bx + c is given by:

Sum of zeroes = -b/a

For the polynomial Kx2 – 3x + 1, a = K, b = -3, and c = 1.

According to the problem, the sum of zeroes is 1:

1 = -(-3)/K → 3/K = 1 → K = 3.

Answer: K = 3

4. Find two numbers where the sum is 27 and the product is 182.

Let the two numbers be x and y. We are given:

x + y = 27 and xy = 182.

Forming the quadratic equation:

t2 – (x + y)t + xy = 0 → t2 – 27t + 182 = 0.

Solving this quadratic equation using the quadratic formula:

t = [27 ± sqrt(729 – 728)]/2 → t = [27 ± 1]/2.

The two values of t are:

t = 14 or t = 13.

Answer: The two numbers are 14 and 13.

5. Formulate a pair of linear equations in two variables: “3 pens and 4 books together cost Rs. 50, whereas 5 pens and 3 books together cost Rs. 54.”

Let the cost of one pen be p and the cost of one book be b. We are given:

3p + 4b = 50

5p + 3b = 54

Answer: The pair of linear equations is:

        3p + 4b = 50
        5p + 3b = 54
6. Verify whether the points (1, 5), (2, 3), (-2, -1) are collinear or not.

We can verify if the points are collinear by checking the area of the triangle formed by them. If the area is 0, the points are collinear.

Using the formula for the area of a triangle formed by the points (x1, y1), (x2, y2), (x3, y3):

        Area = 1/2 * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |

Substituting the points (1, 5), (2, 3), (-2, -1):

        Area = 1/2 * | 1(3 - (-1)) + 2((-1) - 5) + (-2)(5 - 3) |
        = 1/2 * | 1(4) + 2(-6) + (-2)(2) |
        = 1/2 * | 4 - 12 - 4 |
        = 1/2 * |-12| = 6

Since the area is 6 (not 0), the points are not collinear.

Answer: The points are not collinear.

Mathematics Questions and Answers
7. Find the mode of the data: 5, 6, 9, 6, 12, 3, 6, 11, 6, and 7.

The mode is the value that occurs most frequently in a data set.

In the given data set: 5, 6, 9, 6, 12, 3, 6, 11, 6, 7, the number 6 appears 4 times, more than any other number.

Answer: The mode is 6.

8. Express \( \tan \theta \) in terms of \( \sin \theta \).

We know that:

\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Using the identity \( \cos \theta = \sqrt{1 – \sin^2 \theta} \), we get:

Answer: \( \tan \theta = \frac{\sin \theta}{\sqrt{1 – \sin^2 \theta}} \).

9. A doctor observed that the pulse rate of 4 students is 72, 3 students is 78, and 2 students is 80. Find the mean of the pulse rate of the above students.

To find the mean, we use the formula:

Mean = (Sum of all observations) / (Total number of observations)

Answer: The mean pulse rate is approximately 75.78.

10. Find the area of required cloth to cover the heap of grain in conical shape, of whose diameter is 8m and slant height is 3m.

The area of the cloth is the lateral surface area of the cone, which is given by:

Lateral Surface Area = \( \pi r l \), where r is the radius and l is the slant height.

Answer: The area of cloth required is approximately 37.7 m².

11. A die is thrown once. Find the probability of getting an even prime number.

The only even prime number is 2. The total number of outcomes on a die is 6, and the favorable outcome is 2.

Answer: The probability is 1/6.

12. Write the formula for median for grouped data? Explain the symbols in words.

The formula for the median of grouped data is:

Median = \( L + \left( \frac{\frac{N}{2} – CF}{f} \right) \times h \)

Where:

  • L = Lower boundary of the median class
  • N = Total number of observations
  • CF = Cumulative frequency of the class preceding the median class
  • f = Frequency of the median class
  • h = Class width

Answer: The formula for the median of grouped data is as given above.

Math Solutions

Section – II

13. Check whether the given pair of linear equations represent intersecting, parallel or coincident lines. Find the solution if the equations are consistent.

Equations:

1) 3x + 2y = 5

2) 2x – 3y = 5

3) 2x – 3y = 7

4) 4x – 6y = 15

The first pair of equations (3x + 2y = 5 and 2x – 3y = 5) represents intersecting lines, since the determinant of their coefficient matrix is non-zero.

The second pair of equations (2x – 3y = 7 and 4x – 6y = 15) represents parallel lines, as the determinant of their coefficient matrix is zero and they are inconsistent.

14. The number of bacteria in a certain culture triples every hour. If there were 50 bacteria present in the culture originally, what would be the number of bacteria in the 3rd hour? 5th hour? 10th hour? 11th hour?

The number of bacteria triples every hour. The number of bacteria after t hours is given by the formula:

N(t) = 50 * 3^t

For the 3rd hour: N(3) = 50 * 3^3 = 1350 bacteria

For the 5th hour: N(5) = 50 * 3^5 = 12150 bacteria

For the 10th hour: N(10) = 50 * 3^10 = 2952450 bacteria

For the 11th hour: N(11) = 50 * 3^11 = 8857350 bacteria

15. Find the area of the triangle formed by the points (8, –5), (–2, –7), and (5, 1).

The area of a triangle formed by three points (x₁, y₁), (x₂, y₂), (x₃, y₃) is given by:

Area = (1/2) * |x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)|

Substituting the values:

Area = (1/2) * | 8(-7 – 1) + (-2)(1 – (-5)) + 5((-5) – (-7)) | = 33 square units

16. Find the zeros of the quadratic polynomial x² + 5x + 6 and verify the relationship between the zeroes and coefficients.

The zeros of the quadratic polynomial x² + 5x + 6 are found by factoring:

(x + 2)(x + 3) = 0

The zeros are x = -2 and x = -3.

The sum of the zeros is -2 + (-3) = -5, which is the opposite of the coefficient of x. The product of the zeros is (-2)(-3) = 6, which is equal to the constant term.

17. Metallic spheres of radius 6 cm, 8 cm, and 10 cm respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.

The volume of a sphere is given by:

V = (4/3)πr³

The volumes of the individual spheres are added together to form the volume of the resulting sphere:

V₁ + V₂ + V₃ = (4/3)π(6³ + 8³ + 10³)

The radius of the resulting sphere is 12 cm.

18. If secθ + tanθ = P, then find the value of sinθ in terms of P.

We are given:

secθ + tanθ = P

By using trigonometric identities and solving for sinθ, we can derive the value of sinθ in terms of P.

19. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting:

  1. a king of red color
  2. a face card
  3. a red face card
  4. the jack of hearts

We will calculate the probability for each event based on the number of favorable outcomes and the total number of possible outcomes (52 cards).

20. The following table shows marks scored by students in an examination of a certain paper:

Marks 0-10 10-20 20-30 30-40 40-50
Number of Students 20 24 40 36 20

The average marks can be calculated using the deviation method. We first calculate the deviations of class marks from the mean.

Mathematics Solutions

Section – III

21. Draw the graph of P(x) = x² – 6x + 9 and find zeroes. Verify the zeroes of the polynomial.

The given polynomial is P(x) = x² – 6x + 9. This is a quadratic polynomial, and we can factorize it as:

P(x) = (x – 3)(x – 3) = (x – 3)²

The zeroes of the polynomial are x = 3 (a repeated zero). We verify this by substituting x = 3 into P(x), and we get P(3) = 0, confirming that 3 is a zero.

24. Prove that √3 + √5 is an irrational number.

Assume that √3 + √5 is a rational number, i.e., it can be written as a/b where a and b are integers. Then:

√3 + √5 = a/b

Squaring both sides:

3 + 5 + 2√15 = a²/b²

8 + 2√15 = a²/b². Now, subtract 8 from both sides:

2√15 = (a²/b² – 8). This implies that √15 is rational, which is a contradiction since √15 is irrational. Hence, √3 + √5 is irrational.

25. A boat goes 30 km upstream and 44 km downstream in 10 hrs. In 13 hrs it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. Formulate the following problem as a pair of equations and then find the solution.

Let the speed of the boat in still water be x km/h and the speed of the stream be y km/h.

Speed upstream = (x – y) km/h and Speed downstream = (x + y) km/h.

Using the given data, we form the following two equations:

1) (30 / (x – y)) + (44 / (x + y)) = 10

2) (40 / (x – y)) + (55 / (x + y)) = 13

Solving these equations will give the values of x (the speed of the boat in still water) and y (the speed of the stream).

26. Find the coordinates of the points of trisection of the line segment joining the points A(2,-2) and B(-7,4).

The formula for the coordinates of the points dividing a line segment in a ratio m:n is:

Point dividing the segment in the ratio m:n:

((mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n))

Here, A(2, -2) and B(-7, 4), and the line is divided into three equal parts (ratio 1:2).

The coordinates of the first trisection point (P) are:

P = ((1 * -7 + 2 * 2) / 3, (1 * 4 + 2 * -2) / 3) = (-3/3, 0) = (-1, 0)

The coordinates of the second trisection point (Q) are:

Q = ((2 * -7 + 1 * 2) / 3, (2 * 4 + 1 * -2) / 3) = (-12/3, 6/3) = (-4, 2)

27. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle 60°.

This is a geometric construction problem. To draw the tangents:

  • Draw a circle with a radius of 5 cm.
  • From the center of the circle, draw two lines forming a 60° angle between them.
  • Construct the tangents from the points where these lines meet the circle’s circumference.

For accuracy, you can use a protractor and compass for this construction.

28. A right circular cylinder has base radius 14 cm and height 21 cm. Find its:

  1. Area of base (area of each end)
  2. Curved surface area
  3. Total surface area
  4. Volume

The given cylinder has a radius of 14 cm and a height of 21 cm.

1) Area of base = πr² = π * 14² = 616 cm²

2) Curved surface area = 2πrh = 2π * 14 * 21 = 1844 cm²

3) Total surface area = 2πr(h + r) = 2π * 14 * (21 + 14) = 2π * 14 * 35 = 1540 cm²

4) Volume = πr²h = π * 14² * 21 = 6156 cm³

29. If the median of 60 observations, given below is 28.5, find the value of x and y:

Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 5 x 20 15 y 5

We know the formula for the median of grouped data:

Median = L + (N/2 – F)/f * h

Substitute the given values and solve for x and y.

30. Two men on either side of a temple of 30 meter height observe its top at the angles of elevation 30° and 60° respectively. Find the distance between the two men.

Let the distance between the two men be D, and the height of the temple be h = 30 m.

Using trigonometric relations:

tan(30°) = 30/x and tan(60°) = 30/y, where x and y are the distances of each man from the temple.

After calculating x and y using these formulas, the distance between the men is D = x + y.

Multiple Choice Questions

Part – B

  1. One of the following is an irrational number.
    A) 2/3
    B) √(16/25)
    C) √8
    D) √0.04
    Answer: C) √8

    Explanation: √8 is an irrational number because it cannot be expressed as a simple fraction. √(16/25) is a rational number, as it simplifies to 4/5.

  2. The product of zeroes of the cubic polynomial 2x³ – 5x² – 14x + 8 is
    A) -4
    B) 4
    C) -7
    D) 25
    Answer: A) -4

    Explanation: The product of the zeroes of a cubic polynomial ax³ + bx² + cx + d is given by (-d/a). For the given polynomial, d = 8 and a = 2, so the product of the zeroes = -8/2 = -4.

  3. A pair of Linear equations which satisfies dependent system
    A) 2x + y – 5 = 0 ; 3x – 2y – 4 = 0
    B) 3x + 4y = 2 ; 6x + 8y = 4
    C) x + 2y = 3 ; 2x + 4y = 5
    D) x + 2y – 30 = 0 ; 3x + 6y + 60 = 0
    Answer: B) 3x + 4y = 2 ; 6x + 8y = 4

    Explanation: Two linear equations are dependent if one is a multiple of the other. In option B, the second equation is exactly 2 times the first.

  4. The n-th term of G.P. is a_n = ar^(n-1) where ‘r’ represents
    A) First term
    B) Common difference
    C) Common ratio
    D) Radius
    Answer: C) Common ratio

    Explanation: In a geometric progression (G.P.), ‘r’ is the common ratio between successive terms.

  5. The number of two-digit numbers which are divisible by 3
    A) 30
    B) 20
    C) 29
    D) 31
    Answer: A) 30

    Explanation: Two-digit numbers divisible by 3 range from 12 to 99. These numbers form an arithmetic progression with a common difference of 3, and there are 30 such numbers.

  6. The equation of the line which intersects X-axis at (3, 0) is
    A) x + 3 = 0
    B) y + 3 = 0
    C) x – 3 = 0
    D) y – 3 = 0
    Answer: C) x – 3 = 0

    Explanation: The line intersects the X-axis at (3, 0), so the equation of the line is x = 3, or equivalently x – 3 = 0.

  7. The coordinates of the center of the circle if the ends of the diameter are (2, –5) and (–2, 9)
    A) (0, 0)
    B) (2, –2)
    C) (–5, 9)
    D) (0, 2)
    Answer: D) (0, 2)

    Explanation: The center of the circle is the midpoint of the diameter. The midpoint of (2, -5) and (-2, 9) is [(2 + -2)/2, (-5 + 9)/2] = (0, 2).

  8. The point of intersection of the lines x = 2014 and y = 2015 is
    A) (2015, 2014)
    B) (2014, 2015)
    C) (0, 0)
    D) (1, 1)
    Answer: B) (2014, 2015)

    Explanation: The line x = 2014 represents all points where x = 2014, and the line y = 2015 represents all points where y = 2015. The point of intersection is (2014, 2015).

  9. Which of the following vertices form a triangle
    A) (1, 2), (1, 3), (1, 4)
    B) (5, 1), (6, 1), (7, 1)
    C) (0, 0), (–1, 0), (2, 0)
    D) (1, 2), (2, 3), (3, 4)
    Answer: D) (1, 2), (2, 3), (3, 4)

    Explanation: Points are collinear if they lie on the same straight line. In options A, B, and C, the points lie on the same straight line. In option D, the points do not lie on a single straight line and form a triangle.

  10. The slope of a ladder making an angle 30° with the floor
    A) 1
    B) 1/√3
    C) √3
    D) ½
    Answer: B) 1/√3

    Explanation: The slope of the ladder is equal to tan(30°). tan(30°) = 1/√3.